With the material parameters in Fig. REF , the resonant spin waves have wave vector \(k=2\pi /88.9~{\rm nm}^{-1}\) . When taking the Gilbert damping \(\alpha _G=0.02\)  for Ni, the range of spin-wave mediated interaction is \(1/{\rm Im}(q_*)=1.41~\mu {\rm m}\) . The interaction is of long range by choosing the distance of neighboring wires \(L_0=300\)  nm. The chirality is freely tunable by changing the direction of magnetization in the film plane as in Fig. REF . Here we typically choose \(\phi =\lbrace 0.3\pi ,0.54\pi \rbrace \)  that renders \(\Gamma _L/\Gamma _R=0.2\)  and \(\Gamma _R/\Gamma _L=0.2\)  for addressing the physics. In Fig. REF (a), all the modes are localized at the right edge when \(\Gamma _R>\Gamma _L\) , but become localized at the left edge when the chirality is reversed with \(\Gamma _L>\Gamma _R\)  as in Fig. REF (b).
The skin effect vanishes without the chirality at the critical angle \(\phi _c=0.416\pi \) , as shown in Fig. REF (c) that only the modes with large damping have a weak tendacy to be localized at the edge. Also, localization vanishes when taking \(\alpha _G=2\times 10^{-3}\)  [Fig. REF (d)] [1]}. Profoundly, the mode amplitudes are enhanced by two orders in magnitude by the skin effect. This is because that these skin modes are proximity to the \(N\) -th order exceptional points with coalesce of all eigen-vectors [2]}, [3]} when one of \(\Gamma _{L,R}\)  is exactly zero.
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